#283925

order	512
nilpotency class	3
isoclinism	isoclinic class of #264043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#1621	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

(5,6,8,10)(7,9,12,11)(13,14,18,20,17,21,16,22)(15,23,27,24,26,19,25,28);(1,2)(3,4)(6,9)(7,12)(10,11)(13,15,16,25,17,26,18,27)(14,19,22,24,21,23,20,28);(2,4)(6,10)(9,11)(13,14,17,21)(15,24,26,28)(16,20,18,22)(19,27,23,25);(6,9)(7,12)(10,11)(13,16,17,18)(14,20,21,22)(15,25,26,27)(19,28,23,24)

Polycyclic presentation:

2^9|a^2=fi,b^2=fh,c^2=i,d^2=i,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 191121);